GeometryEvolution.cc

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/* Copyright (C) 2016--2025 PISM Authors
 *
 * This file is part of PISM.
 *
 * PISM is free software; you can redistribute it and/or modify it under the
 * terms of the GNU General Public License as published by the Free Software
 * Foundation; either version 3 of the License, or (at your option) any later
 * version.
 *
 * PISM is distributed in the hope that it will be useful, but WITHOUT ANY
 * WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
 * FOR A PARTICULAR PURPOSE.  See the GNU General Public License for more
 * details.
 *
 * You should have received a copy of the GNU General Public License
 * along with PISM; if not, write to the Free Software
 * Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA  02110-1301  USA
*/
 
#include "pism/geometry/GeometryEvolution.hh"
 
#include "pism/util/Grid.hh"
#include "pism/util/Mask.hh"
#include "pism/util/array/CellType.hh"
#include "pism/util/array/Scalar.hh"
#include "pism/util/array/Staggered.hh"
#include "pism/util/array/Vector.hh"
 
#include "pism/geometry/part_grid_threshold_thickness.hh"
#include "pism/util/Context.hh"
#include "pism/util/Logger.hh"
#include "pism/util/Profiling.hh"
#include "pism/util/Interpolation1D.hh"
#include "pism/util/pism_utilities.hh"
 
#include "pism/geometry/flux_limiter.hh"
 
namespace pism {
 
using mask::floating_ice;
using mask::grounded_ice;
using mask::ice_free;
using mask::ice_free_land;
using mask::ice_free_ocean;
using mask::icy;
 
struct GeometryEvolution::Impl {
  Impl(std::shared_ptr<const Grid> g);
 
  GeometryCalculator gc;
 
  double ice_density;
 
  //! True if the basal melt rate contributes to geometry evolution.
  bool use_bmr;
 
  //! True if the part-grid scheme is enabled.
  bool use_part_grid;
 
  //! Flux divergence (used to track thickness changes due to flow).
  array::Scalar flux_divergence;
 
  //! Conservation error due to enforcing non-negativity of ice thickness and enforcing
  //! thickness BC.
  array::Scalar conservation_error;
 
  //! Effective surface mass balance.
  array::Scalar effective_SMB;
 
  //! Effective basal mass balance.
  array::Scalar effective_BMB;
 
  //! Change in ice thickness due to flow during the last time step.
  array::Scalar thickness_change;
 
  //! Change in the ice area-specific volume due to flow during the last time step.
  array::Scalar ice_area_specific_volume_change;
 
  //! Flux through cell interfaces. Ghosted.
  array::Staggered1 flux_staggered;
 
  // Work space
  array::Vector1 input_velocity;       // a ghosted copy; not modified
  array::Scalar1 bed_elevation;        // a copy; not modified
  array::Scalar1 sea_level;            // a copy; not modified
  array::Scalar1 ice_thickness;        // updated in place
  array::Scalar1 area_specific_volume; // updated in place
  array::Scalar1 surface_elevation;    // updated to maintain consistency
  array::CellType1 cell_type;          // updated to maintain consistency
  array::Scalar1 residual;             // temporary storage
  array::Scalar1 thickness;            // temporary storage
};
struct GeometryEvolution::Impl {…};
 
GeometryEvolution::Impl::Impl(std::shared_ptr<const Grid> grid)
    : gc(*grid->ctx()->config()),
      flux_divergence(grid, "flux_divergence"),
      conservation_error(grid, "conservation_error"),
      effective_SMB(grid, "effective_SMB"),
      effective_BMB(grid, "effective_BMB"),
      thickness_change(grid, "thickness_change"),
      ice_area_specific_volume_change(grid, "ice_area_specific_volume_change"),
      flux_staggered(grid, "flux_staggered"),
      input_velocity(grid, "input_velocity"),
      bed_elevation(grid, "bed_elevation"),
      sea_level(grid, "sea_level"),
      ice_thickness(grid, "ice_thickness"),
      area_specific_volume(grid, "area_specific_volume"),
      surface_elevation(grid, "surface_elevation"),
      cell_type(grid, "cell_type"),
      residual(grid, "residual"),
      thickness(grid, "thickness") {
 
  Config::ConstPtr config = grid->ctx()->config();
 
  gc.set_icefree_thickness(config->get_number("geometry.ice_free_thickness_standard"));
 
  // constants
  {
    ice_density   = config->get_number("constants.ice.density");
    use_bmr       = config->get_flag("geometry.update.use_basal_melt_rate");
    use_part_grid = config->get_flag("geometry.part_grid.enabled");
  }
 
  // reported quantities
  {
    // This is the only reported field that is ghosted (we need ghosts to compute flux divergence).
    flux_staggered.metadata(0)
        .long_name("fluxes through cell interfaces (sides) on the staggered grid (x-offset)")
        .units("m^2 s^-1")
        .output_units("m^2 year^-1");
    flux_staggered.metadata(1)
        .long_name("fluxes through cell interfaces (sides) on the staggered grid (y-offset)")
        .units("m^2 s^-1")
        .output_units("m^2 year^-1");
 
    flux_divergence.metadata(0)
        .long_name("flux divergence")
        .units("m s^-1")
        .output_units("m year^-1");
 
    conservation_error.metadata(0)
        .long_name(
            "conservation error due to enforcing non-negativity of ice thickness (over the last time step)")
        .units("meters");
 
    effective_SMB.metadata(0)
        .long_name("effective surface mass balance over the last time step")
        .units("meters");
 
    effective_BMB.metadata(0)
        .long_name("effective basal mass balance over the last time step")
        .units("meters");
 
    thickness_change.metadata(0).long_name("change in thickness due to flow").units("meters");
 
    ice_area_specific_volume_change.metadata(0)
        .long_name("change in area-specific volume due to flow")
        .units("meters^3 / meters^2");
  }
 
  // internal storage
  {
    input_velocity.metadata(0)
        .long_name("ghosted copy of the input velocity")
        .units("meters / second");
 
    bed_elevation.metadata(0).long_name("ghosted copy of the bed elevation").units("meters");
 
    sea_level.metadata(0).long_name("ghosted copy of the sea level elevation").units("meters");
 
    ice_thickness.metadata(0)
        .long_name("working (ghosted) copy of the ice thickness")
        .units("meters");
 
    area_specific_volume.metadata(0)
        .long_name("working (ghosted) copy of the area specific volume")
        .units("meters^3 / meters^2");
 
    surface_elevation.metadata(0)
        .long_name("working (ghosted) copy of the surface elevation")
        .units("meters");
 
    cell_type.metadata(0).long_name("working (ghosted) copy of the cell type mask");
 
    residual.metadata(0).long_name("residual area specific volume").units("meters^3 / meters^2");
 
    thickness.metadata(0).long_name("thickness (temporary storage)").units("meters");
  }
}
GeometryEvolution::Impl::Impl(std::shared_ptr<const Grid> grid) {…}
 
GeometryEvolution::GeometryEvolution(std::shared_ptr<const Grid> grid) : Component(grid) {
  m_impl = new Impl(grid);
}
GeometryEvolution::GeometryEvolution(std::shared_ptr<const Grid> grid) : Component(grid) {…}
 
GeometryEvolution::~GeometryEvolution() {
  delete m_impl;
}
GeometryEvolution::~GeometryEvolution() {…}
 
void GeometryEvolution::init(const InputOptions &opts) {
  this->init_impl(opts);
}
void GeometryEvolution::init(const InputOptions &opts) {…}
 
void GeometryEvolution::init_impl(const InputOptions &opts) {
  (void)opts;
  // empty: the default implementation has no state
}
void GeometryEvolution::init_impl(const InputOptions &opts) {…}
 
void GeometryEvolution::reset() {
  m_impl->conservation_error.set(0.0);
}
void GeometryEvolution::reset() {…}
 
const array::Scalar &GeometryEvolution::flux_divergence() const {
  return m_impl->flux_divergence;
}
const array::Scalar &GeometryEvolution::flux_divergence() const  {…}
 
const array::Staggered1 &GeometryEvolution::flux_staggered() const {
  return m_impl->flux_staggered;
}
const array::Staggered1 &GeometryEvolution::flux_staggered() const  {…}
 
const array::Scalar &GeometryEvolution::top_surface_mass_balance() const {
  return m_impl->effective_SMB;
}
const array::Scalar &GeometryEvolution::top_surface_mass_balance() const  {…}
 
const array::Scalar &GeometryEvolution::bottom_surface_mass_balance() const {
  return m_impl->effective_BMB;
}
const array::Scalar &GeometryEvolution::bottom_surface_mass_balance() const  {…}
 
const array::Scalar &GeometryEvolution::thickness_change_due_to_flow() const {
  return m_impl->thickness_change;
}
const array::Scalar &GeometryEvolution::thickness_change_due_to_flow() const  {…}
 
const array::Scalar &GeometryEvolution::area_specific_volume_change_due_to_flow() const {
  return m_impl->ice_area_specific_volume_change;
}
const array::Scalar &GeometryEvolution::area_specific_volume_change_due_to_flow() const  {…}
 
const array::Scalar &GeometryEvolution::conservation_error() const {
  return m_impl->conservation_error;
}
const array::Scalar &GeometryEvolution::conservation_error() const  {…}
 
/*!
 * @param[in] geometry ice geometry
 * @param[in] dt time step, seconds
 * @param[in] advective_velocity advective (SSA) velocity
 * @param[in] diffusive_flux diffusive (SIA) flux
 * @param[in] velocity_bc_values advective velocity Dirichlet B.C. values
 * @param[in] thickness_bc_mask ice thickness Dirichlet B.C. mask
 *
 * Results are stored in internal fields accessible using getters.
 */
void GeometryEvolution::flow_step(const Geometry &geometry, double dt,
                                  const array::Vector &advective_velocity,
                                  const array::Staggered &diffusive_flux,
                                  const array::Scalar &thickness_bc_mask) {
 
  profiling().begin("ge.update_ghosted_copies");
  {
    // make ghosted copies of input fields
    m_impl->ice_thickness.copy_from(geometry.ice_thickness);
    m_impl->area_specific_volume.copy_from(geometry.ice_area_specific_volume);
    m_impl->sea_level.copy_from(geometry.sea_level_elevation);
    m_impl->bed_elevation.copy_from(geometry.bed_elevation);
    m_impl->input_velocity.copy_from(advective_velocity);
 
    // Compute cell_type and surface_elevation. Ghosts of results are updated.
    m_impl->gc.compute(m_impl->sea_level,          // in (uses ghosts)
                       m_impl->bed_elevation,      // in (uses ghosts)
                       m_impl->ice_thickness,      // in (uses ghosts)
                       m_impl->cell_type,          // out (ghosts are updated)
                       m_impl->surface_elevation); // out (ghosts are updated)
  }
  profiling().end("ge.update_ghosted_copies");
 
  // Derived classes can include modifications for regional runs.
  profiling().begin("ge.interface_fluxes");
  compute_interface_fluxes(m_impl->cell_type,       // in (uses ghosts)
                           m_impl->ice_thickness,   // in (uses ghosts)
                           m_impl->input_velocity,  // in (uses ghosts)
                           diffusive_flux,          // in
                           m_impl->flux_staggered); // out
  profiling().end("ge.interface_fluxes");
 
  m_impl->flux_staggered.update_ghosts();
 
  {
    // allocate temporary storage (FIXME: at some point I should evaluate whether it's OK
    // to allocate this every time step)
    array::Staggered flux_limited(m_grid, "limited_ice_flux");
 
    make_nonnegative_preserving(dt,
                                m_impl->ice_thickness,  // in (uses ghosts)
                                m_impl->flux_staggered, // in (uses ghosts)
                                flux_limited);
 
    m_impl->flux_staggered.copy_from(flux_limited);
  }
 
  profiling().begin("ge.flux_divergence");
  compute_flux_divergence(dt,                         // in
                          m_impl->flux_staggered,     // in (uses ghosts)
                          thickness_bc_mask,          // in
                          m_impl->conservation_error, // in/out
                          m_impl->flux_divergence);   // out
  profiling().end("ge.flux_divergence");
 
  // This is where part_grid is implemented.
  profiling().begin("ge.update_in_place");
  update_in_place(dt,                            // in
                  m_impl->bed_elevation,         // in
                  m_impl->sea_level,             // in
                  m_impl->flux_divergence,       // in
                  m_impl->ice_thickness,         // in/out
                  m_impl->area_specific_volume); // in/out
  profiling().end("ge.update_in_place");
 
  // Compute ice thickness and area specific volume changes.
  profiling().begin("ge.compute_changes");
  {
    m_impl->ice_thickness.add(-1.0, geometry.ice_thickness, m_impl->thickness_change);
    m_impl->area_specific_volume.add(-1.0, geometry.ice_area_specific_volume,
                                     m_impl->ice_area_specific_volume_change);
  }
  profiling().end("ge.compute_changes");
 
  // Computes the numerical conservation error and corrects ice_thickness_change and
  // ice_area_specific_volume_change. We can do this here because
  // compute_surface_and_basal_mass_balance() preserves non-negativity.
  //
  // Note that here we use the "old" ice geometry.
  //
  // This computation is purely local.
  profiling().begin("ge.ensure_nonnegativity");
  ensure_nonnegativity(geometry.ice_thickness,                  // in
                       geometry.ice_area_specific_volume,       // in
                       m_impl->thickness_change,                // in/out
                       m_impl->ice_area_specific_volume_change, // in/out
                       m_impl->conservation_error);             // in/out
  profiling().end("ge.ensure_nonnegativity");
 
  // Now the caller can compute
  //
  // H_new    = H_old + thickness_change
  // Href_new = Href_old + ice_area_specific_volume_change.
 
  // calving is a separate issue
}
void GeometryEvolution::flow_step(const Geometry &geometry, double dt, {…}
 
void GeometryEvolution::source_term_step(const Geometry &geometry, double dt,
                                         const array::Scalar &thickness_bc_mask,
                                         const array::Scalar &surface_mass_balance_rate,
                                         const array::Scalar &basal_melt_rate) {
 
  profiling().begin("ge.source_terms");
  compute_surface_and_basal_mass_balance(dt,                        // in
                                         thickness_bc_mask,         // in
                                         geometry.ice_thickness,    // in
                                         geometry.cell_type,        // in
                                         surface_mass_balance_rate, // in
                                         basal_melt_rate,           // in
                                         m_impl->effective_SMB,     // out
                                         m_impl->effective_BMB);    // out
  profiling().end("ge.source_terms");
}
void GeometryEvolution::source_term_step(const Geometry &geometry, double dt, {…}
 
/*!
 * Apply changes due to flow to ice geometry and ice area specific volume.
 */
void GeometryEvolution::apply_flux_divergence(Geometry &geometry) const {
  geometry.ice_thickness.add(1.0, m_impl->thickness_change);
  geometry.ice_area_specific_volume.add(1.0, m_impl->ice_area_specific_volume_change);
}
void GeometryEvolution::apply_flux_divergence(Geometry &geometry) const  {…}
 
/*!
 * Update geometry by applying changes due to surface and basal mass fluxes.
 *
 * Note: This method performs these changes in the same order as the code ensuring
 * non-negativity. This is important.
 */
void GeometryEvolution::apply_mass_fluxes(Geometry &geometry) const {
 
  const array::Scalar &dH_SMB = top_surface_mass_balance(), &dH_BMB = bottom_surface_mass_balance();
  array::Scalar &H = geometry.ice_thickness;
 
  array::AccessScope list{ &H, &dH_SMB, &dH_BMB };
  ParallelSection loop(m_grid->com);
  try {
    for (auto p = m_grid->points(); p; p.next()) {
      const int i = p.i(), j = p.j();
 
      // To preserve non-negativity of thickness we need to apply changes in this exact order.
      // (Recall that floating-point arithmetic is not associative.)
      const double H_new = (H(i, j) + dH_SMB(i, j)) + dH_BMB(i, j);
 
#if (Pism_DEBUG == 1)
      if (H_new < 0.0) {
        throw RuntimeError::formatted(PISM_ERROR_LOCATION, "H = %f (negative) at i=%d, j=%d", H_new,
                                      i, j);
      }
#endif
 
      H(i, j) = H_new;
    }
  } catch (...) {
    loop.failed();
  }
  loop.check();
}
void GeometryEvolution::apply_mass_fluxes(Geometry &geometry) const  {…}
 
 
/*!
 * Prevent advective ice flow from floating ice to ice-free land, as well as in the
 * ice-free areas.
 *
 * Note: positive `input` corresponds to the flux from `current` to `neighbor`.
 */
static double limit_advective_flux(int current, int neighbor, double input) {
 
  // No flow from ice-free ocean:
  if ((ice_free_ocean(current) and input > 0.0) or (ice_free_ocean(neighbor) and input < 0.0)) {
    return 0.0;
  }
 
  // No flow from ice-free land:
  if ((ice_free_land(current) and input > 0.0) or (ice_free_land(neighbor) and input < 0.0)) {
    return 0.0;
  }
 
  // Case 1: Flow between grounded_ice and grounded_ice.
  if (grounded_ice(current) and grounded_ice(neighbor)) {
    return input;
  }
 
  // Cases 2 and 3: Flow between grounded_ice and floating_ice.
  if ((grounded_ice(current) and floating_ice(neighbor)) or
      (floating_ice(current) and grounded_ice(neighbor))) {
    return input;
  }
 
  // Cases 4 and 5: Flow between grounded_ice and ice_free_land.
  if ((grounded_ice(current) and ice_free_land(neighbor)) or
      (ice_free_land(current) and grounded_ice(neighbor))) {
    return input;
  }
 
  // Cases 6 and 7: Flow between grounded_ice and ice_free_ocean.
  if ((grounded_ice(current) and ice_free_ocean(neighbor)) or
      (ice_free_ocean(current) and grounded_ice(neighbor))) {
    return input;
  }
 
  // Case 8: Flow between floating_ice and floating_ice.
  if (floating_ice(current) and floating_ice(neighbor)) {
    return input;
  }
 
  // Cases 9 and 10: Flow between floating_ice and ice_free_land.
  if ((floating_ice(current) and ice_free_land(neighbor)) or
      (ice_free_land(current) and floating_ice(neighbor))) {
    // Disable all flow. This ensures that an ice shelf does not climb up a cliff.
    return 0.0;
  }
 
  // Cases 11 and 12: Flow between floating_ice and ice_free_ocean.
  if ((floating_ice(current) and ice_free_ocean(neighbor)) or
      (ice_free_ocean(current) and floating_ice(neighbor))) {
    return input;
  }
 
  // Case 13: Flow between ice_free_land and ice_free_land.
  if (ice_free_land(current) and ice_free_land(neighbor)) {
    return 0.0;
  }
 
  // Cases 14 and 15: Flow between ice_free_land and ice_free_ocean.
  if ((ice_free_land(current) and ice_free_ocean(neighbor)) or
      (ice_free_ocean(current) and ice_free_land(neighbor))) {
    return 0.0;
  }
 
  // Case 16: Flow between ice_free_ocean and ice_free_ocean.
  if (ice_free_ocean(current) and ice_free_ocean(neighbor)) {
    return 0.0;
  }
 
  throw RuntimeError::formatted(
      PISM_ERROR_LOCATION, "cannot handle the case current=%d, neighbor=%d", current, neighbor);
}
static double limit_advective_flux(int current, int neighbor, double input) {…}
 
/*!
 * Prevent SIA-driven flow in ice shelves and ice-free areas.
 *
 * Note: positive `flux` corresponds to the flux from `current` to `neighbor`.
 */
static double limit_diffusive_flux(int current, int neighbor, double flux) {
 
  // No flow from ice-free ocean:
  if ((ice_free_ocean(current) and flux > 0.0) or (ice_free_ocean(neighbor) and flux < 0.0)) {
    return 0.0;
  }
 
  // No flow from ice-free land:
  if ((ice_free_land(current) and flux > 0.0) or (ice_free_land(neighbor) and flux < 0.0)) {
    return 0.0;
  }
 
  // Case 1: Flow between grounded_ice and grounded_ice.
  if (grounded_ice(current) and grounded_ice(neighbor)) {
    return flux;
  }
 
  // Cases 2 and 3: Flow between grounded_ice and floating_ice.
  if ((grounded_ice(current) and floating_ice(neighbor)) or
      (floating_ice(current) and grounded_ice(neighbor))) {
    return flux;
  }
 
  // Cases 4 and 5: Flow between grounded_ice and ice_free_land.
  if ((grounded_ice(current) and ice_free_land(neighbor)) or
      (ice_free_land(current) and grounded_ice(neighbor))) {
    return flux;
  }
 
  // Cases 6 and 7: Flow between grounded_ice and ice_free_ocean.
  if ((grounded_ice(current) and ice_free_ocean(neighbor)) or
      (ice_free_ocean(current) and grounded_ice(neighbor))) {
    return flux;
  }
 
  // Case 8: Flow between floating_ice and floating_ice.
  if (floating_ice(current) and floating_ice(neighbor)) {
    // no diffusive flux in ice shelves
    return 0.0;
  }
 
  // Cases 9 and 10: Flow between floating_ice and ice_free_land.
  if ((floating_ice(current) and ice_free_land(neighbor)) or
      (ice_free_land(current) and floating_ice(neighbor))) {
    // Disable all flow. This ensures that an ice shelf does not climb up a cliff.
    return 0.0;
  }
 
  // Cases 11 and 12: Flow between floating_ice and ice_free_ocean.
  if ((floating_ice(current) and ice_free_ocean(neighbor)) or
      (ice_free_ocean(current) and floating_ice(neighbor))) {
    return 0.0;
  }
 
  // Case 13: Flow between ice_free_land and ice_free_land.
  if (ice_free_land(current) and ice_free_land(neighbor)) {
    return 0.0;
  }
 
  // Cases 14 and 15: Flow between ice_free_land and ice_free_ocean.
  if ((ice_free_land(current) and ice_free_ocean(neighbor)) or
      (ice_free_ocean(current) and ice_free_land(neighbor))) {
    return 0.0;
  }
 
  // Case 16: Flow between ice_free_ocean and ice_free_ocean.
  if (ice_free_ocean(current) and ice_free_ocean(neighbor)) {
    return 0.0;
  }
 
  throw RuntimeError::formatted(
      PISM_ERROR_LOCATION, "cannot handle the case current=%d, neighbor=%d", current, neighbor);
}
static double limit_diffusive_flux(int current, int neighbor, double flux) {…}
 
/*!
 * Combine advective velocity and the diffusive flux on the staggered grid with the ice thickness to
 * compute the total flux through cell interfaces.
 *
 * Uses first-order upwinding to compute the advective flux.
 *
 * Limits the diffusive flux to prevent SIA-driven flow in the ocean and ice-free areas.
 */
void GeometryEvolution::compute_interface_fluxes(const array::CellType1 &cell_type,
                                                 const array::Scalar &ice_thickness,
                                                 const array::Vector &velocity,
                                                 const array::Staggered &diffusive_flux,
                                                 array::Staggered &output) {
 
  array::AccessScope list{ &cell_type, &velocity, &ice_thickness, &diffusive_flux, &output };
 
  ParallelSection loop(m_grid->com);
  try {
    // compute advective fluxes and put them in output
    for (auto p = m_grid->points(); p; p.next()) {
      const int i = p.i(), j = p.j(), M = cell_type.as_int(i, j);
 
      const double H   = ice_thickness(i, j);
      const Vector2d& V = velocity(i, j);
 
      for (int n = 0; n < 2; ++n) {
        const int oi = 1 - n,  // offset in the i direction
            oj       = n,      // offset in the j direction
            i_n      = i + oi, // i index of a neighbor
            j_n      = j + oj; // j index of a neighbor
 
        const int M_n = cell_type.as_int(i_n, j_n);
 
        // advective velocity at the current interface
        double v = 0.0;
        {
          const Vector2d& V_n = velocity(i_n, j_n);
          int W = static_cast<int>(icy(M)), W_n = static_cast<int>(icy(M_n));
 
          auto v_staggered = (W * V + W_n * V_n) / std::max(W + W_n, 1);
          v                = n == 0 ? v_staggered.u : v_staggered.v;
        }
 
        // advective flux
        const double H_n         = ice_thickness(i_n, j_n),
                     Q_advective = v * (v > 0.0 ? H : H_n); // first order upwinding
 
        output(i, j, n) = Q_advective;
      } // end of the loop over neighbors (n)
    }
 
    // limit the advective flux and add the diffusive flux to it to get the total
    for (auto p = m_grid->points(); p; p.next()) {
      const int i = p.i(), j = p.j(), M = cell_type.as_int(i, j);
 
      for (int n = 0; n < 2; ++n) {
        const int oi = 1 - n,  // offset in the i direction
            oj       = n,      // offset in the j direction
            i_n      = i + oi, // i index of a neighbor
            j_n      = j + oj; // j index of a neighbor
 
        const int M_n = cell_type.as_int(i_n, j_n);
 
        // diffusive flux
        const double Q_diffusive = limit_diffusive_flux(M, M_n, diffusive_flux(i, j, n)),
                     Q_advective = limit_advective_flux(M, M_n, output(i, j, n));
 
        output(i, j, n) = Q_diffusive + Q_advective;
      } // end of the loop over n
    }
 
  } catch (...) {
    loop.failed();
  }
  loop.check();
}
void GeometryEvolution::compute_interface_fluxes(const array::CellType1 &cell_type, {…}
 
/*!
 * Compute flux divergence using cell interface fluxes on the staggered grid.
 *
 * The flux divergence at *ice thickness* Dirichlet B.C. locations is set to zero.
 */
void GeometryEvolution::compute_flux_divergence(double dt, const array::Staggered1 &flux,
                                                const array::Scalar &thickness_bc_mask,
                                                array::Scalar &conservation_error,
                                                array::Scalar &output) {
  const double dx = m_grid->dx(), dy = m_grid->dy();
 
  array::AccessScope list{ &flux, &thickness_bc_mask, &conservation_error, &output };
 
  ParallelSection loop(m_grid->com);
  try {
    for (auto p = m_grid->points(); p; p.next()) {
      const int i = p.i(), j = p.j();
 
      auto Q = flux.star(i, j);
 
      double divQ = (Q.e - Q.w) / dx + (Q.n - Q.s) / dy;
 
      if (thickness_bc_mask(i, j) > 0.5) {
        // the thickness change would have been equal to -divQ*dt. By keeping ice
        // thickness fixed we *add* divQ*dt meters of ice.
        conservation_error(i, j) += divQ * dt; // units: meters
        output(i, j) = 0.0;
      } else {
        output(i, j) = divQ;
      }
    }
  } catch (...) {
    loop.failed();
  }
  loop.check();
}
void GeometryEvolution::compute_flux_divergence(double dt, const array::Staggered1 &flux, {…}
 
/*!
 * Update ice thickness and area_specific_volume *in place*.
 *
 * It would be better to compute the change in ice thickness and area_specific_volume and then apply
 * them, but it would require re-writing all the part-grid code from scratch. So, I make copies of
 * ice thickness and area_specific_volume, use this old code, then compute differences to get changes.
 * Compute ice thickness changes due to the flow of the ice.
 *
 * @param[in] dt time step, seconds
 * @param[in] bed_elevation bed elevation, meters
 * @param[in] sea_level sea level elevation
 * @param[in] ice_thickness ice thickness
 * @param[in] area_specific_volume area-specific volume (m3/m2)
 * @param[in] flux_divergence flux divergence
 * @param[out] thickness_change ice thickness change due to flow
 * @param[out] area_specific_volume_change area specific volume change due to flow
 */
void GeometryEvolution::update_in_place(double dt, const array::Scalar &bed_topography,
                                        const array::Scalar &sea_level,
                                        const array::Scalar &flux_divergence,
                                        array::Scalar &ice_thickness,
                                        array::Scalar &area_specific_volume) {
 
  m_impl->gc.compute(sea_level, bed_topography, ice_thickness, m_impl->cell_type,
                     m_impl->surface_elevation);
 
  array::AccessScope list{ &ice_thickness, &flux_divergence };
 
  if (m_impl->use_part_grid) {
    m_impl->residual.set(0.0);
 
    // Store ice thickness. We need this copy to make sure that modifying ice_thickness in the loop
    // below does not affect the computation of the threshold thickness. (Note that
    // part_grid_threshold_thickness uses neighboring values of the mask, ice thickness, and surface
    // elevation.)
    m_impl->thickness.copy_from(ice_thickness);
 
    list.add({ &area_specific_volume, &m_impl->residual, &m_impl->thickness,
               &m_impl->surface_elevation, &bed_topography, &m_impl->cell_type });
  }
 
  const double Lz = m_grid->Lz();
 
  ParallelSection loop(m_grid->com);
  try {
    for (auto p = m_grid->points(); p; p.next()) {
      const int i = p.i(), j = p.j();
 
      double divQ = flux_divergence(i, j);
 
      if (m_impl->use_part_grid) {
        if (m_impl->cell_type.ice_free_ocean(i, j) and m_impl->cell_type.next_to_ice(i, j)) {
          assert(divQ <= 0.0);
          // Add the flow contribution to this partially filled cell.
          area_specific_volume(i, j) += -divQ * dt;
 
          double threshold = part_grid_threshold_thickness(
              m_impl->cell_type.star_int(i, j), m_impl->thickness.star(i, j),
              m_impl->surface_elevation.star(i, j), bed_topography(i, j));
 
          // if threshold is zero, turn all the area specific volume into ice thickness, with zero
          // residual
          if (threshold == 0.0) {
            threshold = area_specific_volume(i, j);
          }
 
          if (area_specific_volume(i, j) >= threshold) {
            ice_thickness(i, j) += threshold;
            m_impl->residual(i, j)     = area_specific_volume(i, j) - threshold;
            area_specific_volume(i, j) = 0.0;
          }
 
          // In this case the flux goes into the area_specific_volume variable and does not directly
          // contribute to ice thickness at this location.
          divQ = 0.0;
        }
      } // end of if (use_part_grid)
 
      ice_thickness(i, j) += -dt * divQ;
 
      if (ice_thickness(i, j) > Lz) {
        throw RuntimeError::formatted(PISM_ERROR_LOCATION,
                                      "ice thickness would exceed Lz at i=%d, j=%d (H=%f, Lz=%f)",
                                      i, j, ice_thickness(i, j), Lz);
      }
    }
  } catch (...) {
    loop.failed();
  }
  loop.check();
 
  ice_thickness.update_ghosts();
 
  // Compute the mask corresponding to the new thickness.
  m_impl->gc.compute_mask(sea_level, bed_topography, ice_thickness, m_impl->cell_type);
 
  /*
    Redistribute residual ice mass from subgrid-scale parameterization.
 
    See [@ref Albrechtetal2011].
  */
  if (m_impl->use_part_grid) {
    const int max_n_iterations = (int)m_config->get_number("geometry.part_grid.max_iterations");
 
    bool done = false;
    for (int i = 0; i < max_n_iterations and not done; ++i) {
      m_log->message(4, "redistribution iteration %d\n", i);
 
      // this call may set done to true
      residual_redistribution_iteration(bed_topography, sea_level, m_impl->surface_elevation,
                                        ice_thickness, m_impl->cell_type, area_specific_volume,
                                        m_impl->residual, done);
    }
 
    if (not done) {
      m_log->message(
          2,
          "WARNING: not done redistributing mass after %d iterations, remaining residual: %f m^3.\n",
          max_n_iterations, sum(m_impl->residual) * m_grid->cell_area());
 
      // Add residual to ice thickness, preserving total ice mass. (This is not great, but
      // better than losing mass.)
      ice_thickness.add(1.0, m_impl->residual);
      m_impl->residual.set(0.0);
    }
 
    if (max(ice_thickness) > m_grid->Lz()) {
      throw RuntimeError::formatted(PISM_ERROR_LOCATION, "ice thickness would exceed Lz "
                                                         "after part_grid residual redistribution");
    }
  }
}
void GeometryEvolution::update_in_place(double dt, const array::Scalar &bed_topography, {…}
 
//! @brief Perform one iteration of the residual mass redistribution.
/*!
  @param[in] bed_topography bed elevation
  @param[in] sea_level sea level elevation
  @param[in,out] ice_surface_elevation surface elevation; used as temp. storage
  @param[in,out] ice_thickness ice thickness; updated
  @param[in,out] cell_type cell type mask; used as temp. storage
  @param[in,out] area_specific_volume area specific volume; updated
  @param[in,out] residual ice volume that still needs to be distributed; updated
  @param[in,out] done result flag: true if this iteration should be the last one
 */
void GeometryEvolution::residual_redistribution_iteration(const array::Scalar &bed_topography,
                                                          const array::Scalar &sea_level,
                                                          array::Scalar1 &ice_surface_elevation,
                                                          array::Scalar &ice_thickness,
                                                          array::CellType1 &cell_type,
                                                          array::Scalar &area_specific_volume,
                                                          array::Scalar &residual, bool &done) {
 
  m_impl->gc.compute_mask(sea_level, bed_topography, ice_thickness, cell_type);
 
  // First step: distribute residual mass
  {
    // will be destroyed at the end of the block
    array::AccessScope list{ &cell_type, &ice_thickness, &area_specific_volume, &residual };
 
    for (auto p = m_grid->points(); p; p.next()) {
      const int i = p.i(), j = p.j();
 
      if (residual(i, j) <= 0.0) {
        continue;
      }
 
      auto m = cell_type.star_int(i, j);
 
      int N = 0; // number of empty or partially filled neighbors
      for (auto d : { North, East, South, West }) {
        N += ice_free_ocean(m[d]) ? 1 : 0;
      }
 
      if (N > 0) {
        // Remaining ice mass will be redistributed equally among all adjacent
        // ice-free-ocean cells (is there a more physical way?)
        residual(i, j) /= N;
      } else {
        // Conserve mass, but (possibly) create a "ridge" at the shelf
        // front
        ice_thickness(i, j) += residual(i, j);
        residual(i, j) = 0.0;
      }
    }
 
    residual.update_ghosts();
 
    // update area_specific_volume using adjusted residuals
    for (auto p = m_grid->points(); p; p.next()) {
      const int i = p.i(), j = p.j();
 
      if (cell_type.ice_free_ocean(i, j)) {
        area_specific_volume(i, j) +=
            (residual(i + 1, j) + residual(i - 1, j) + residual(i, j + 1) + residual(i, j - 1));
      }
    }
 
    residual.set(0.0);
  }
 
  ice_thickness.update_ghosts();
 
  // Store ice thickness. We need this copy to make sure that modifying ice_thickness in the loop
  // below does not affect the computation of the threshold thickness. (Note that
  // part_grid_threshold_thickness uses neighboring values of the mask, ice thickness, and surface
  // elevation.)
  m_impl->thickness.copy_from(ice_thickness);
 
  // The loop above updated ice_thickness, so we need to re-calculate the mask and the
  // surface elevation:
  m_impl->gc.compute(sea_level, bed_topography, ice_thickness, cell_type, ice_surface_elevation);
 
  double remaining_residual = 0.0;
 
  // Second step: we need to redistribute residual ice volume if
  // neighbors which gained redistributed ice also become full.
  {
    // will be destroyed at the end of the block
    array::AccessScope list{ &m_impl->thickness, &ice_thickness, &ice_surface_elevation,
                             &bed_topography, &cell_type };
 
    for (auto p = m_grid->points(); p; p.next()) {
      const int i = p.i(), j = p.j();
 
      if (area_specific_volume(i, j) <= 0.0) {
        continue;
      }
 
      double threshold =
          part_grid_threshold_thickness(cell_type.star_int(i, j), m_impl->thickness.star(i, j),
                                        ice_surface_elevation.star(i, j), bed_topography(i, j));
 
      // if threshold is zero, turn all the area specific volume into ice thickness, with zero
      // residual
      if (threshold == 0.0) {
        threshold = area_specific_volume(i, j);
      }
 
      if (area_specific_volume(i, j) >= threshold) {
        ice_thickness(i, j) += threshold;
        residual(i, j)             = area_specific_volume(i, j) - threshold;
        area_specific_volume(i, j) = 0.0;
 
        remaining_residual += residual(i, j);
      }
    }
  }
 
  // check if redistribution should be run once more
  remaining_residual = GlobalSum(m_grid->com, remaining_residual);
 
  done = remaining_residual <= 0.0;
 
  ice_thickness.update_ghosts();
}
void GeometryEvolution::residual_redistribution_iteration(const array::Scalar &bed_topography, {…}
 
/*!
 * Correct `thickness_change` and `area_specific_volume_change` so that applying them will not
 * result in negative `ice_thickness` and `area_specific_volume`.
 *
 * Compute the the amount of ice that is added to preserve non-negativity of ice thickness.
 *
 * @param[in] ice_thickness ice thickness (m)
 * @param[in] area_specific_volume area-specific volume (m3/m2)
 * @param[in,out] thickness_change "proposed" thickness change (m)
 * @param[in,out] area_specific_volume_change "proposed" area-specific volume change (m3/m2)
 * @param[in,out] conservation_error computed conservation error (m)
 *
 * This computation is purely local.
 */
void GeometryEvolution::ensure_nonnegativity(const array::Scalar &ice_thickness,
                                             const array::Scalar &area_specific_volume,
                                             array::Scalar &thickness_change,
                                             array::Scalar &area_specific_volume_change,
                                             array::Scalar &conservation_error) {
 
  array::AccessScope list{ &ice_thickness, &area_specific_volume, &thickness_change,
                           &area_specific_volume_change, &conservation_error };
 
  ParallelSection loop(m_grid->com);
  try {
    for (auto p = m_grid->points(); p; p.next()) {
      const int i = p.i(), j = p.j();
 
      const double H = ice_thickness(i, j), dH = thickness_change(i, j);
 
      // applying thickness_change will lead to negative thickness
      if (H + dH < 0.0) {
        thickness_change(i, j) = -H;
        conservation_error(i, j) += -(H + dH);
      }
 
      const double V = area_specific_volume(i, j), dV = area_specific_volume_change(i, j);
 
      if (V + dV < 0.0) {
        area_specific_volume_change(i, j) = -V;
        conservation_error(i, j) += -(V + dV);
      }
    }
  } catch (...) {
    loop.failed();
  }
  loop.check();
}
void GeometryEvolution::ensure_nonnegativity(const array::Scalar &ice_thickness, {…}
 
/*!
 * Given ice thickness `H` and the "proposed" change `dH`, compute the corrected change preserving
 * non-negativity.
 */
static inline double effective_change(double H, double dH) {
  if (H + dH <= 0) {
    return -H;
  }
  return dH;
}
static inline double effective_change(double H, double dH) {…}
 
/*!
 * Compute effective surface and basal mass balance.
 *
 * @param[in] dt time step, seconds
 * @param[in] thickness_bc_mask mask specifying ice thickness Dirichlet B.C. locations
 * @param[in] ice_thickness ice thickness, m
 * @param[in] thickness_change thickness change due to flow, m
 * @param[in] cell_type cell type mask
 * @param[in] smb_rate top surface mass balance rate, kg m-2 s-1
 * @param[in] basal_melt_rate basal melt rate, m s-1
 * @param[out] effective_smb effective top surface mass balance, m
 * @param[out] effective_bmb effective basal mass balance, m
 *
 * This computation is purely local.
 *
 * Positive outputs correspond to mass gain.
 */
void GeometryEvolution::compute_surface_and_basal_mass_balance(
    double dt, const array::Scalar &thickness_bc_mask, const array::Scalar &ice_thickness,
    const array::CellType &cell_type, const array::Scalar &surface_mass_flux,
    const array::Scalar &basal_melt_rate, array::Scalar &effective_SMB,
    array::Scalar &effective_BMB) {
 
  array::AccessScope list{ &ice_thickness,     &surface_mass_flux,      &basal_melt_rate, &cell_type,
                           &thickness_bc_mask, &effective_SMB, &effective_BMB };
 
  ParallelSection loop(m_grid->com);
  try {
    for (auto p = m_grid->points(); p; p.next()) {
      const int i = p.i(), j = p.j();
 
      // Don't modify ice thickness at Dirichlet B.C. locations and in the ice-free ocean.
      if (thickness_bc_mask.as_int(i, j) == 1 or cell_type.ice_free_ocean(i, j)) {
        effective_SMB(i, j) = 0.0;
        effective_BMB(i, j) = 0.0;
        continue;
      }
 
      const double H = ice_thickness(i, j);
 
      // Thickness change due to the surface mass balance
      //
      // Note that here we convert surface mass balance from [kg m-2 s-1] to [m s-1].
      double dH_SMB = effective_change(H, dt * surface_mass_flux(i, j) / m_impl->ice_density);
 
      // Thickness change due to the basal mass balance
      //
      // Note that basal_melt_rate is in [m s-1]. Here the negative sign converts the melt rate into
      // mass balance.
      double dH_BMB =
          effective_change(H + dH_SMB, dt * (m_impl->use_bmr ? -basal_melt_rate(i, j) : 0.0));
 
      effective_SMB(i, j) = dH_SMB;
      effective_BMB(i, j) = dH_BMB;
    }
  } catch (...) {
    loop.failed();
  }
  loop.check();
}
void GeometryEvolution::compute_surface_and_basal_mass_balance( {…}
 
namespace diagnostics {
 
/*! @brief Report the divergence of the ice flux. */
class FluxDivergence : public Diag<GeometryEvolution> {
public:
  FluxDivergence(const GeometryEvolution *m) : Diag<GeometryEvolution>(m) {
    m_vars = { model->flux_divergence().metadata() };
  }
  FluxDivergence(const GeometryEvolution *m) : Diag<GeometryEvolution>(m) {…}
 
protected:
  std::shared_ptr<array::Array> compute_impl() const {
    auto result = allocate<array::Scalar>("flux_divergence");
 
    result->copy_from(model->flux_divergence());
 
    return result;
  }
  std::shared_ptr<array::Array> compute_impl() const  {…}
};
class FluxDivergence : public Diag<GeometryEvolution> {…};
 
/*! @brief Report mass flux on the staggered grid. */
class FluxStaggered : public Diag<GeometryEvolution> {
public:
  FluxStaggered(const GeometryEvolution *m) : Diag<GeometryEvolution>(m) {
    m_vars = { model->flux_staggered().metadata(0), model->flux_staggered().metadata(1) };
  }
  FluxStaggered(const GeometryEvolution *m) : Diag<GeometryEvolution>(m) {…}
 
protected:
  std::shared_ptr<array::Array> compute_impl() const {
    auto result = allocate<array::Staggered>("flux_staggered");
 
    result->copy_from(model->flux_staggered());
 
    return result;
  }
  std::shared_ptr<array::Array> compute_impl() const  {…}
};
class FluxStaggered : public Diag<GeometryEvolution> {…};
 
} // end of namespace diagnostics
 
DiagnosticList GeometryEvolution::diagnostics_impl() const {
  using namespace diagnostics;
  typedef Diagnostic::Ptr Ptr;
 
  std::map<std::string, Ptr> result;
  result = {
    { "flux_staggered", Ptr(new FluxStaggered(this)) },
    { "flux_divergence", Ptr(new FluxDivergence(this)) },
  };
  return result;
}
DiagnosticList GeometryEvolution::diagnostics_impl() const  {…}
 
RegionalGeometryEvolution::RegionalGeometryEvolution(std::shared_ptr<const Grid> grid)
    : GeometryEvolution(grid), m_no_model_mask(m_grid, "no_model_mask") {
 
  m_no_model_mask.set_interpolation_type(NEAREST);
 
  m_no_model_mask.metadata(0).long_name("'no model' mask");
}
RegionalGeometryEvolution::RegionalGeometryEvolution(std::shared_ptr<const Grid> grid) {…}
 
void RegionalGeometryEvolution::set_no_model_mask_impl(const array::Scalar &mask) {
  m_no_model_mask.copy_from(mask);
}
void RegionalGeometryEvolution::set_no_model_mask_impl(const array::Scalar &mask) {…}
 
/*!
 * Disable ice flow in "no model" areas.
 */
void RegionalGeometryEvolution::compute_interface_fluxes(const array::CellType1 &cell_type,
                                                         const array::Scalar        &ice_thickness,
                                                         const array::Vector        &velocity,
                                                         const array::Staggered     &diffusive_flux,
                                                         array::Staggered           &output) {
 
  GeometryEvolution::compute_interface_fluxes(cell_type, ice_thickness,
                                              velocity, diffusive_flux,
                                              output);
 
  array::AccessScope list{&m_no_model_mask, &output};
 
  ParallelSection loop(m_grid->com);
  try {
    for (auto p = m_grid->points(); p; p.next()) {
      const int i = p.i(), j = p.j();
 
      const int M = m_no_model_mask.as_int(i, j);
 
      for (int n : {0, 1}) {
        const int
          oi  = 1 - n,               // offset in the i direction
          oj  = n,                   // offset in the j direction
          i_n = i + oi,              // i index of a neighbor
          j_n = j + oj;              // j index of a neighbor
 
        const int M_n = m_no_model_mask.as_int(i_n, j_n);
 
        if (not (M == 0 and M_n == 0)) {
          output(i, j, n) = 0.0;
        }
      }
    }
  } catch (...) {
    loop.failed();
  }
  loop.check();
}
void RegionalGeometryEvolution::compute_interface_fluxes(const array::CellType1 &cell_type, {…}
 
/*!
 * Set surface and basal mass balance to zero in "no model" areas.
 */
void RegionalGeometryEvolution::compute_surface_and_basal_mass_balance(double dt,
                                                                       const array::Scalar      &thickness_bc_mask,
                                                                       const array::Scalar        &ice_thickness,
                                                                       const array::CellType &cell_type,
                                                                       const array::Scalar        &surface_mass_flux,
                                                                       const array::Scalar        &basal_melt_rate,
                                                                       array::Scalar              &effective_SMB,
                                                                       array::Scalar              &effective_BMB) {
  GeometryEvolution::compute_surface_and_basal_mass_balance(dt,
                                                            thickness_bc_mask,
                                                            ice_thickness,
                                                            cell_type,
                                                            surface_mass_flux,
                                                            basal_melt_rate,
                                                            effective_SMB,
                                                            effective_BMB);
 
  array::AccessScope list{&m_no_model_mask, &effective_SMB, &effective_BMB};
 
  ParallelSection loop(m_grid->com);
  try {
    for (auto p = m_grid->points(); p; p.next()) {
      const int i = p.i(), j = p.j();
 
      if (m_no_model_mask(i, j) > 0.5) {
        effective_SMB(i, j) = 0.0;
        effective_BMB(i, j) = 0.0;
      }
    }
  } catch (...) {
    loop.failed();
  }
  loop.check();
}
void RegionalGeometryEvolution::compute_surface_and_basal_mass_balance(double dt, {…}
 
void GeometryEvolution::set_no_model_mask(const array::Scalar &mask) {
  this->set_no_model_mask_impl(mask);
}
void GeometryEvolution::set_no_model_mask(const array::Scalar &mask) {…}
 
void GeometryEvolution::set_no_model_mask_impl(const array::Scalar &mask) {
  (void) mask;
  // the default implementation is a no-op
}
void GeometryEvolution::set_no_model_mask_impl(const array::Scalar &mask) {…}
/*!
 * Return the volumetric ice flow rate from land to water (across the grounding line), in
 * m^3 / s. Positive values correspond to ice moving from the grounded side to the
 * floating (ocean) side.
 *
 * Mass continuity equation without source terms:
 *
 * dH/dt = - div(Q)
 *
 * Approximating the flux divergence, we get
 *
 * - div(Q) ~= - ((Q.e - Q.w) / dx + (Q.n - Q.s) / dy)
 *
 * Multiplying by the cell area we get the volume flow rate
 *
 * dV/dt = - dx *dy * div(Q)
 *
 * which can be approximated by
 *
 * - (dy * (Q.e - Q.w) + dx * (Q.n - Q.s)) = dy * (Q.w - Q.e) + dx * (Q.s - Q.n)
 */
static double volume_flow_rate_from_land_to_water(const stencils::Star<int> &cell_type,
                                                  const stencils::Star<double> &flux, double dx,
                                                  double dy) {
  using mask::grounded;
 
  auto Q = flux; // units: m^2 / s
 
  // zero out fluxes between the current (floating or ocean) cell and other (floating or
  // ocean) cells
  Q.n *= (int)grounded(cell_type.n);
  Q.e *= (int)grounded(cell_type.e);
  Q.s *= (int)grounded(cell_type.s);
  Q.w *= (int)grounded(cell_type.w);
 
  return dy * (Q.w - Q.e) + dx * (Q.s - Q.n); // units: m^3 / s
}
static double volume_flow_rate_from_land_to_water(const stencils::Star<int> &cell_type, {…}
 
/*!
 * Compute the ice flow rate across the grounding line, adding to `output` to accumulate
 * contributions from multiple time steps.
 *
 * When `unit_conversion_factor` is 1 the units of `output` are "m^3 / s".
 *
 * Negative flux corresponds to ice moving into the ocean, i.e. from grounded to floating
 * areas. (This convention makes it easier to compare this quantity to the surface mass
 * balance or calving fluxes.)
 *
 * Different choices of the `unit_conversion_factor` make it possible to use this in
 *
 * - the grounding line flux diagnostic (in kg / (m^2 s)),
 * - the volume flow rate diagnostic (in m^3 / s),
 * - or the mass flow rate diagnostic (in kg / s).
 */
void ice_flow_rate_across_grounding_line(const array::CellType1 &cell_type,
                                         const array::Staggered1 &flux,
                                         double unit_conversion_factor, array::Scalar &output) {
  auto grid = output.grid();
 
  const double dx = grid->dx(), // units: m
      dy          = grid->dy(); // units: m
 
  array::AccessScope list{ &cell_type, &flux, &output };
 
  ParallelSection loop(grid->com);
  try {
    for (auto p = grid->points(); p; p.next()) {
      const int i = p.i(), j = p.j();
 
      double result = 0.0;
 
      if (cell_type.ocean(i, j)) {
        auto M = cell_type.star_int(i, j);
        auto Q = flux.star(i, j); // units: m^2 / s
 
        // note the sign change: here *negative* values correspond to ice moving from
        // grounded to floating areas since it can sometimes be interpreted as "mass loss"
        result = -volume_flow_rate_from_land_to_water(M, Q, dx, dy); // units: m^3 / s
 
        // convert from "m^3 / s" to "kg"
        result *= unit_conversion_factor;
      }
 
      output(i, j) += result;
    }
  } catch (...) {
    loop.failed();
  }
  loop.check();
}
void ice_flow_rate_across_grounding_line(const array::CellType1 &cell_type, {…}
 
double total_grounding_line_flux(const array::CellType1 &cell_type,
                                 const array::Staggered1 &flux,
                                 double dt) {
  auto grid = cell_type.grid();
 
  const double
    dx = grid->dx(),            // units: m
    dy = grid->dy();            // units: m
 
  auto ice_density = grid->ctx()->config()->get_number("constants.ice.density"); // units: kg / m^3
 
  double conversion_factor = dt * ice_density;
 
  double total_flux = 0.0;
 
  array::AccessScope list{&cell_type, &flux};
 
  ParallelSection loop(grid->com);
  try {
    for (auto p = grid->points(); p; p.next()) {
      const int i = p.i(), j = p.j();
 
      if (cell_type.ocean(i, j)) {
        auto M = cell_type.star_int(i, j);
        auto Q = flux.star(i, j); // units: m^2 / s
 
        // note the sign change: here *negative* values correspond to ice moving from
        // grounded to floating areas since it can sometimes be interpreted as "mass loss"
        double volume_flux = -volume_flow_rate_from_land_to_water(M, Q, dx, dy); // units: m^3 / s
        // convert from "m^3 / s" to "kg" and sum up
        total_flux += volume_flux * conversion_factor;
      }
    }
  } catch (...) {
    loop.failed();
  }
  loop.check();
 
  return GlobalSum(grid->com, total_flux);
}
double total_grounding_line_flux(const array::CellType1 &cell_type, {…}
 
} // end of namespace pism